# (square roots of the identity matrix) For how many 2x2 matrices A is it true that A^2=I ? Now answer the same question for n x n matrices where n>2

(square roots of the identity matrix) For how many $2×2$ matrices A is it true that ${A}^{2}=I$ ? Now answer the same question for $n×x$ n matrices where $n>2$

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Step 1
To find how many matricec A of order 2 exists such that ${A}^{2}=I$
Here we have to find ${A}^{2}=A×A=I$
Let us consider any arbitrary matrix of order 2. $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$
Then ${A}^{2}$ is given by ${A}^{2}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}{a}^{2}+bc& ab+bd\\ ca+dc& {d}^{2}+bc\end{array}\right)$
Now from ${A}^{2}=I$ , we get ${A}^{2}=\left(\begin{array}{cc}{a}^{2}+bc& b\left(a+d\right)\\ c\left(a+d\right)& {d}^{2}+bc\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=I$
$\left(1\right){a}^{2}+bc=1$
$\left(2\right){d}^{2}+bc=1$
$\left(3\right)b\left(a+d\right)=0$
$\left(3\right)c\left(a+d\right)=0$
Step 2
Case 1:
If $\left(a+d\right)\ne 0$
From (3) we get : $b=0$
From (4) we get : $c=0$
From (1) we get : ${a}^{2}=1⇒a=±1$
From (2) we get : ${d}^{2}=1⇒d=±1$
Therefore we get the matrix of the form
$\left\{A=\left(\begin{array}{cc}a& 0\\ 0& d\end{array}\right):a=±1,d=±1\right\}$ Therefore there are only 4 matrices exist such that ${A}^{2}=I$ from case 1.
Step 3
Case 2:
If $\left(a+d\right)=0$
sub case a:
If a=0 , that implies $d=0$
Then from(1 ) and (2) , we get : $bc=1⇒c={b}^{-1}=\frac{1}{b}$ in real numbers.
Therefore we get the matrix of the form
$\left\{A=\left(\begin{array}{cc}0& a\\ {a}^{-1}& 0\end{array}\right):a\in F\right\}$ Therefore there are infinitely many matrices exist if the field is the set of real numbers such that ${A}^{2}=I$
Step 4
We can also consider further subcases to find the type of involuntary matrices, but as the question is asked to find how many, we have already got infinitely many $2×2$ matrices of such type.
Similarly for any $n>2$ , we can find infinitely many matrices satisfying this condition on the field of real numbers.

Jeffrey Jordon
Jeffrey Jordon